{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<h3>Regularised regression</h3>\n",
    "<p>Our original squared loss function in matrix/vector notation is:\n",
    "$$ L = \\frac{1}{N}\\sum_{n=1}^N (\\mathbf{t} - \\mathbf{X}\\mathbf{w})^T(\\mathbf{t} - \\mathbf{X}\\mathbf{w}) $$\n",
    "Here's another loss function:\n",
    "$$ L = \\lambda \\mathbf{w}^T\\mathbf{w} + \\frac{1}{N}\\sum_{n=1}^N (\\mathbf{t} - \\mathbf{X}\\mathbf{w})^T(\\mathbf{t} - \\mathbf{X}\\mathbf{w}) $$\n",
    "Recall that we're minimising this function and so (if $\\lambda>0$) this additional term will penalise large positive and negative values in $\\mathbf{w}$. $\\lambda$ controls how much influence this new term has over the original squared error term.</p>\n",
    "\n",
    "<p>Differentiating this with respect to $\\mathbf{w}$ and then setting to zero (this is a good exercise to do) results in:\n",
    "$$ (\\mathbf{X}^T\\mathbf{X} + \\lambda\\mathbf{I})\\mathbf{w} = \\mathbf{X}^T\\mathbf{t} $$\n",
    "where $\\mathbf{I}$ is a square matrix with ones on the diagonal and zeros elsewhere (the identity matrix).</p>\n",
    "\n",
    "<p>To demonstrate the effect of this additional term, we will generate some synthetic data by using a quadratic function and assing some random (normal / Gaussian) noise.</p>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x10c14b390>]"
      ]
     },
     "execution_count": 1,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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PP2b+/PkqAyLnoRECEQkaK1eu5LHHHiMnJ4fBgwfzwgsvaBqhyAUq0wiBMWa4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\nRU5ErkJggg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x10c14b310>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "import numpy as np\n",
    "x = np.linspace(0,10,10)[:,None]\n",
    "y = x**2 - 3*x + 4\n",
    "t = y + np.random.normal(0,10,(x.size,1))\n",
    "import pylab as plt\n",
    "%matplotlib inline\n",
    "plt.plot(x,y,'k')\n",
    "plt.plot(x,t,'ro')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<h4>Creating $\\mathbf{X}$</h4>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "maxorder = 7\n",
    "x_test = np.linspace(0,10,30)[:,None]\n",
    "X = np.ones_like(x)\n",
    "X_test = np.ones_like(x_test)\n",
    "for i in range(1,maxorder+1):\n",
    "    X = np.hstack((X,x**i))\n",
    "    X_test = np.hstack((X_test,x_test**i))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<h4>Loop over different values of $\\lambda$</h4>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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      "text/plain": [
       "<matplotlib.figure.Figure at 0x10c5ba110>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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f/wB/J8Arr8QDxXvvxY9dvhwefdQ/QiHfI9GvH5xwAnTtqg/AxoTD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\nSZoChIiIiCTt/wOgoTFjdzBOnwAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x10ce8f050>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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QIpIW8vLymFJRwa0jRjB+6lTa19WxIjeXPgMHMmXUqPU+wrlkiV8k6umn4dVX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\nH1ojpgQvvlutAAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x10d542790>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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XLPmwPbnkPgUIEQncu+/6IYo//Qm++KLxazvtBOecA5dd5p+eyIXP01zbnlzy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\nDcCeTdo7s3mvRCOVlZV0bPL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      "text/plain": [
       "<matplotlib.figure.Figure at 0x10d4b6f50>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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mzmXG/PkKBZLSFAhEJG0sWuSHBj74ILz8cvPn7LKLnzHw+OPh4IP9DILJMLGk\nhFHRRZrqGTA4EsHV1DCptNRP0SySohQIRCSlLV3qZwt88EG/XHCD6+2PCgp8CDjhBD9RUBCP7udV\nVjKuueLwoWDyzJmgQCApTIFARFLOihUwcyZUVPiOgeHwuufsvbefMvjkk/1dgSA55+gSDtNSDjGg\nczisjoaS0hQIRCQlrFoFs2b5OwGVlc2PDthlFxg+3L/23jv5NbbEzKjNzcVBs6HAAbW5uQoDktIU\nCEQkMHV18Mwz/k7Ao4/C8uXrnrP99v4uwPDhsP/+qbuKYL8hQ5hTXt6oD0G92Tk59B86NICqRDae\nAoGIJN2iRXD33XDnnX7IYFNbbuknCioqgv79IScn+TW21ugJExg2dy4u2rGwfpTB7JwcpuTnM2P8\n+KBLFFkvBQIRSQrn4Pnn4fbb/d2AtWsbH+/WzXcKHD4cBg6E3NxAytxkeXl5zJg/n0mlpUyeOZPO\n4TArc3PpN3QoM8aP15BDSXkKBCKSUEuXwn33wV/+Au+/3/iYmZ8s6Pe/95MFJXKegGTIy8vzQwvL\nytSBUNKOAoGIxJ1zsGCBDwEPPug7DDbUs6cPAeecA336BFNjoikMSLpRIBCRuFmxAh54wAeB119f\n9/ihh8L55/tHAx06JL8+EWmZAoGItNnbb/u+AfffD6FQ42Pdu8OZZ8J550F+fjD1iciGKRCIyCaJ\nRPx8ARMnwksvrXv8gAN8CBg+HDp3Tn59ItI6CgQi0ip1dX49geuug//8p/Gxzp3hlFN8ECgsDKY+\nEdk0CgQislHCYZg2Da6/Hj78sPGxvfbyfQNOOy0xywqLSOIpEIjIeq1aBffcAzfd5Jccbujgg6G0\n1A8ZVKd6kfSmQCAizVqxAu64w/cRWLSo8bEBA3wQGDBAQUAkUygQiEgj338Pf/4z3HILLFnS+NjR\nR0NJCfziF8HUJiKJ06oZws1sjJktMLPlZrbYzB41s92bnNPRzMrN7FszC5nZw2bWM75li0i8ffON\nv9jvtBNcdVUsDJjBsGFQVQVPPKEwIJKpWrtkyC+BW4GDgMOBXOApM9uswTm3AMcAw4BDgO2AGW0v\nVUQS4csvYdQo6N3bjxyoX3EwJ8d3Enz7bXj4YSgoCLRMEUmwVj0ycM4d3XDfzH4LfA0UAi+ZWTfg\nd8Bw59wL0XPOAmrM7EDn3IK4VC0ibfbddzB+PJSXw5o1sfbcXPjtb+GKK2CXXQIrT0SSrK19CDbH\nr/D5XXR3Q9jcAAATlElEQVS/MPqez9af4Jx738wWAn0BBQKRgK1a5fsITJjg+wvU69QJ/vAHGD0a\nevUKrj4RCcYmBwLzK3fcArzknHs32rwNsMY5t7zJ6Yujx0Qkjlqzol4kAhUVvp9Aw+GDnTrByJH+\nscHWWyeoUBFJeW25Q3AbsBfQfyPONfydhBYVFxfTvXv3Rm1FRUUUFRVtcoEimSgUCjGxpIR5lZV0\nCYepzc2l35AhjJ4wgby8vGZ/5rnn/Df/6upYm5l/NHDNNbDDDsmpXUQ2XUVFBRUVFY3ali1bFrf3\nN+fWe51u/ofM/gwMAX7pnFvYoH0A8AywRcO7BGb2KTDFOVfWzHsVAFVVVVUUqNeSyHqFQiGG9e3L\nqJoaBkUiPybtOTk5TM7PZ8b8+Y1CwTvv+L4ATzzR+H0GDfITDf3850ktX0TirLq6mkI/T3ihc656\nQ+evT2tHGdSHgeOAAQ3DQFQVsBYY2OD83YEdgfltqFNEgIklJYyqqWFwNAyAv/02OBKhuKaGSaWl\nAHz1FZxzjr/gNwwD++4LTz0Fs2crDIhIY616ZGBmtwFFwFCg1szqnzguc86tcs4tN7O7gclmthQI\nAVOBeRphINJ28yorGReJNHtscCTCxMdmMnbzMiZOhJUrY8d22MF3IjztND+cUESkqdb2ITgPf4fy\n+SbtZwF/j24XA3XAw0BHYDZw4aaXKCLgOxB2CYdpqQuhAaHPwlxzjYvuQbduMGYMXHwxbLZZCz8o\nIkLr5yHY4HcL59xq4KLoS0TixMyozc0ldrlvzAFLIrmA0b49XHCBn3GwR4/k1iki6Uk3D0XSSL8h\nQ5jTwj3/x8lhCUM58USoqYGyMoUBEdl4CgQiaWT0hAlM3COfxy3nx3G8Dqgkh+LN8nn46fE89BDs\numuQVYpIOtJqhyJpwjl4/PE83lk6n9NcKT2YSXfC1LbPpfDIoVRXjKdbt+bnIRAR2RAFApE08Pbb\nMGIEvPACQB5QxqqOZZxyuWPMGFOHQRFpMwUCkRS2bBmMGwe33gp1dbH2IUNgyhTYZZeNm7ZYRGRD\nFAhEUpBzMG0aXHYZLF4ca995Z5g6FY45JrjaRCQzKRCIpJg334QLL4R582JtnTrBH//oA0KnTsHV\nJiKZS4FAJEV8/72fN+C22/zKhPVOOAEmT4bevQMrTUSygAKBSMAiEbj3XrjySvjmm1j7brv5vgOD\nBgVWmohkEQUCkQC9/rqfUfCVV2JtnTtDaSmMGgUdOwZXm4hkFwUCkQAsWwZ/+hP8+c+NHw+ceCJM\nmgQ77hhcbSKSnRQIRJLIOfjHP/y3/6++irXvuad/PHD44cHVJiLZTVMXiyTJBx/AkUdCUVEsDGy2\nGVx/vR9ZoDAgIkHSHQKRBPvhB3/Rv/FGWLMm1j5kiJ9TQKMHRCQVKBCIJNDs2X7K4Y8/jrXtuKN/\nPDB0aHB1iYg0pUcGIgnwxRdw0klw1FGxMNC+PVxxBbz7rsKAiKQe3SEQiaO1a/1jgLFjYcWKWPsh\nh/gJh3760+BqExFZHwUCkTh5+WU4/3x4661Y21ZbwcSJcPrpYFqHSERSmB4ZiLTRkiVwzjnQr18s\nDJjBuefCe+/BGWcoDIhI6tMdApFNFInA3/7mpxz+9ttY+777wl/+AgcdFFxtIiKtpTsEIptgwQI4\n+GA4++xYGMjLg7Iy+Pe/FQZEJP3oDoFIK3z9NYwZA/fc07j95JP9ioTbbRdMXSIibaVAILIR1q71\nowT+9Ce/DkG9n/7UzykwYEBwtYmIxIMCgcgGPP88XHQRvP12rK1bN7jmGr9SYW5uYKWJiMSNAoFI\nCz7/HEaP9osRNXTWWX4q4q23DqYuEZFEUCAQaWL1at8fYPx4WLky1r7//n65YnUYFJFMpEAg0sAT\nT8All8BHH8XaevTwdwR+9zvI0bgcEclQ+udNBB8AhgyBY4+NhYGcHL8w0Qcf+OGFCgMiksl0h0A2\nyDmHZehUe7W1/tv/zTc3Xpr4l7/0owf22Se42kREkknfeaRZoVCIsSNHcnifPhzfqxeH9+nD2JEj\nCYVCQZcWF+Ew3H477LorTJgQCwPbbQcPPAAvvKAwICLZRXcIZB2hUIhhffsyqqaGcZEIBjhgTnk5\nw+bOZcb8+eTl5QVd5iaJRGD6dCgtjS1LDH7o4KhRvr1r1+DqExEJiu4QyDomlpQwqqaGwdEwAGDA\n4EiE4poaJpWWBlneJnEOnnrKjxQoKmocBoYN83MM3HCDwoCIZC8FAlnHvMpKBkUizR4bHIkwb+bM\nJFfUNgsWwMCBMGgQvP56rH3AAHj1VXj4Ydh99+DqExFJBQoE0ohzji7hMC11ITSgcziMcy6ZZW2S\n996DE0/08wY891ysfb/9YM4cePZZOPDA4OoTEUklCgTSiJlRm5tLS5d7B9Tm5qb0qIMvvoBzzoG9\n94YZM2Ltu+wCFRXw2mtw5JGQwn8FEZGkUyCQdfQbMoQ5LQy6n52TQ/+hQ5Nc0cZZuhSuuMKPHLjr\nLqir8+1bb+0XJqqpgeHDNZ+AiEhzNMpA1jF6wgSGzZ2La9Cx0OHDwJT8fGaMHx90iY2sXAlTp8KN\nN8L338fau3WDyy/3Mw926RJcfSIi6UCBQNaRl5fHjPnzmVRayuSZM+kcDrMyN5d+Q4cyY/z4lBly\nGArBPffATTfBl1/G2jt29DMMjhkDW24ZXH0iIulEgSABMmFmv7y8PMaVlUFZWcr9fT77zM8ieOed\nsGxZrD0nB848E8aNgx13DKw8EZG0pEAQJ6FQiIklJcyrrKRLOExtbi79hgxh9IQJKfONelOlShh4\n7TW/CuH06bH+AfWOP97POLjXXsHUJiKS7hQI4iCTZ/YLWl0dVFb6IPDii42PdegAp5/u+wjsvXcw\n9YmIZIqU6W993rHHpu1c+Zk4s1/QamuhvBz23BNOOKFxGOjRA8aOhYUL/WgChQERkbZLmUBw+1df\n0be8nGF9+6ZdKMi0mf2C9MUXvjNgr16+Y2D9UsTgw8Gdd/ogMG6cH04oIiLxkTKBIF2/UWfSzH5B\nev11f/u/d2+/psDSpbFjAwfCk0/CO+/4CYc22yywMkVEMlbKBIJ66faNOhNm9gtKKORnDhwwAAoK\nYNo0WLvWH8vN9SMG3ngDnnkGjjpKEwqJiCRSynUqbPiNOl0uov2GDGFOeTmDm3lskMoz+wWhthYe\nf9yPFHjySVi1qvHxn/wEzj8fLrwQtt02mBpFRLJRygWCdPxGnW4z+yXbypX+4j99ug8DP/yw7jm7\n7QbFxXDGGZpVUEQkCCkXCNLxG3W6zOyXTD/8ALNm+RBQWelDQVM9e8KwYXDSSXDooXokICISJAu6\ns5uZFQBVrwFf13+jTvNx++n0uCOeVq2C2bNjIWDFinXP6dHDh4Df/AYOOQTap1wkFRFJH9XV1RQW\nFgIUOueq2/JeKfPP8QXbbstRJ52UEd+osykMrFgBzz3nQ8Bjj/mOgk1tuSX8+tc+BPzqVwoBIiKp\nKGX+ab798ccpKCgIugxZj0gEPvgAXnkl9vrPf3x7U1tsEQsBAwb4UQMiIpK6EhYIzOxCYDSwDfAm\ncJFz7t+J+n0Sf999BwsWxC7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      "text/plain": [
       "<matplotlib.figure.Figure at 0x10ca34a10>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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XAROdcxUAZnYOsAIYAjyxvYWKiIjUFY36RwXV1X7/t7+Fgw4KtqZM1txHBkXAO2b2hJmt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\nfz0KEBFJZ2kXCDLxG3XfoiL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      "text/plain": [
       "<matplotlib.figure.Figure at 0x10d4a2c90>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "for lamb in [0,0.01,0.1,1,10,100]:\n",
    "    w = np.linalg.solve(np.dot(X.T,X) + lamb*np.identity(maxorder+1),np.dot(X.T,t))\n",
    "    f_test = np.dot(X_test,w)\n",
    "    plt.figure()\n",
    "    plt.plot(x_test,f_test,'b-',linewidth=2)\n",
    "    plt.plot(x,t,'ro')\n",
    "    title = '$\\lambda=$%g'%lamb\n",
    "    plt.title(title)\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<p>As $\\lambda$ increases, high values in $\\mathbf{w}$ are more heavily penalised which leads to *simpler* functions. Why do lower values correspond to simpler functions?</p>\n",
    "<p>Firstly, what does *simpler* mean?</p>\n",
    "<p>I would argue that simpler functions have smaller derivatives (first, second, etc) as they typically change more slowly. In our polynomials, the derivatives are dependent on the values of $\\mathbf{w}$. In particular our polynomial is:\n",
    "$$ t = \\sum_{d=0}^D w_d x^d $$\n",
    "and the first derivative is:\n",
    "$$ \\frac{dt}{dx} = \\sum_{d=1}^D dw_d x^{d-1} $$\n",
    "and second is:\n",
    "$$ \\frac{d^2t}{dx^2} = \\sum_{d=2}^D d(d-1)w_d x^{d-2} $$\n",
    "which in both cases increases with increasing values of $w_d$. So penalising high (positive and negative) values decreases (in general) the gradients (and gradients of gradients, etc).</p> "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": []
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 2",
   "language": "python",
   "name": "python2"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 2
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython2",
   "version": "2.7.10"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 0
}
